Why do I only breathe out of one nostril?

This is a natural phenomenon called the nasal cycle. It is discussed in this paper by Telles et al. (1994), among many others. The nostrils are used on an alternating cycle of about 2-3 hours, controlled by the autonomic nervous system. If you notice alternating congestion, that also seems to be coupled to the nasal cycle (Hasegawa and Kern 1977, 1978).

Normally, the direct cause is blockage caused by alternating engorgement of erectile tissues on each side of the nose.

Below, I explain that the nasal cycle may be an artifact of the way the autonomous system works in humans (and some other organisms) or it may provide a first barrier against invasion of infective organisms via the nose. The full answer requires some background information.

Wikipedia and the nasal cycle

The Wikipedia article on the nasal cycle, nicely paraphrased in the answer by George Daccache, offers hints but not real answers. For example, the Wiki section called Research on the effects states,

In 1994, breathing through alternate nostrils showed effects on brain hemisphere symmetry on EEG topography.

suggesting that perhaps natural nasal cycling relates somehow to communication or coordination between the two brain hemispheres. However, the cited study (Stancak and Kuna 1994) is based on forced alternate nostril breathing (FANB). The Wiki section then states,

D.S. Shannahoff-Khalsa published in 2007 on the effect of this cycle and manipulation through forced nostril breathing on one side on the endogenous ultradian rhythms of the autonomic and central nervous system.

All this sentence ultimately says is that a 2007 paper looked at the effects of FANB on ultradian rhythms. The Wiki section doesn't actually include results from the study so it provides no useful information.$^1$ I will try to explain with more depth.

What is Forced Air Nostril Breathing?

Forced Air Nostril Breathing (FANB) requires a person to close one nostril, breath in, close the second nostril and open the first nostril, and breath out. The person repeats this process several times in a 10-15 minute period. In fact, FANB is a yoga technique called Nadi Shodhan Pranayama. The Telles et al. paper mentioned above used FANB, as has nearly every study on the effects of the nasal cycle.

None of these studies actually explain the natural nasal cycle. FANB is not natural. It changes natural breathing rhythms and requires the person to focus on the physical movements of the fingers to close the alternating nostrils. FANB occurs for a 10-15 minute period of time and is finished. This is not the same as the 2-3 hour natural nasal cycle. In my opinion, any conclusions drawn from studies using FANB apply only to FANB but not to the natural nasal cycle.

Then why does the nasal cycle occur?

The nasal cycle is a natural ultradian cycle (see here and here. Not only is it present in humans, but the nasal cycle has also been observed in rats, rabbits, domestic pigs, cats and dogs (see references in Eccles 1996]). Thus, the nasal cycle may at least be a feature of mammals but it may be a feature of other bilateral animals that use nostrils for respiration. In addition, the nasal cycle may be an artifact of the evolution of bilateral symmetry in animals, and how the autonomic nervous system operates between the two sides.

The autonomic nervous system controls the nasal cycle. The autonomic nervous system has two divisions, the sympathetic nervous system and the parasympathetic nervous system. Interestingly, these two divisions show a lateralized ultradian rhythm (Shannahoff-Khalsa 2007). This means that the parasympathetic nervous system dominates one side of the body and the sympathetic nervous system dominates the other side of the body. The two systems later switch dominate sides. This dominance switching back and forth between the parasympathetic and sympathetic happens with a regular rhythmic cycle every few hours. As it happens, this switching between sides correlates very well with the nasal cycle (Shannahoff-Khalsa 1991). When the parasympathetic-sympathetic systems switch sides, so do the nostrils. This is also associated with a switch in EEG activity between the two brain hemispheres (Werntz et al. 1983).

Therefore, the nasal cycle may not have a specific function, adaptive or otherwise. Instead, it could result from dominance of the parasympathetic system. Whichever side is dominated by the parasympathetic system will have the primary nostril in use for respiration. However, others have argued that the nasal cycle does provide a function. For example, Eccles (1996) argued that the nasal cycle may function as a respiratory defense mechanism. They found that the rate of cycling increases when nasal infection is present in the nose. They argue that the congestion-decongestion helps generate "plasma exudate" (nasal fluids derived from blood plasma) which may help remove bacteria and viruses.

The nasal cycle is an interesting phenomenon but whether it evolved as an adaptation (such as a mechanism proposed by Eccles et al. (1996) or is simply an artifact of the operation of the autonomic nervous system may never be known for sure.

Footnote

1. One of many reasons why you should always interpret Wikipedia entries very cautiously, even skeptically.

Citations

Eccles, R. 1996. A role for the nasal cycle in respiratory defense. European Respiratory Journal 9: 371-376.

Hasegawa, M. and E.B. Kern. 1977. The human nasal cycle. Mayo Clinic Proceedings 52: 28-34.

Hasegawa, M. and E.B. Kern. 1978. Variations in nasal resistance in man: a rhinomanometric study of the nasal cycle in 50 human subjects. Rhinology 16: 19-29.

Stancak, A. and M. Kuna. 1994. EEG changes during forced alternate nostril breathing. International Journal of Psychophysiology 16: 75-79.

Telles, S. et al. 1994. Breathing through a particular nostril can alter metabolism and autonomic activities.

Werntz, D.A. et al. 1983. Alternating cerebral hemispheric activity and the lateralization of autonomic nervous function. Human Neurobiology 2: 39-43.

As others have said, this phenomena is called the nasal cycle, a process controlled by the autonomic nervous system that alternants congestion between your nostrils. Mentalfloss of all places has an article about this that explains:

...it makes our sense of smell more complete. Different scent molecules degrade at different rates, and our scent receptors pick up on them accordingly. Some smells are easier to detect and process in a fast-moving airstream like the decongested nostril, while others are better detected in the slower airstream of the congested nostril. Nasal cycling also seems to keep the nose maintained for its function as an air filter and humidifier. The alternating congestion gives the mucous and cilia (the tiny hairs up in your nose) in each nostril a well-deserved break from the onslaught of air and prevents the insides of your nostrils from drying out, cracking and bleeding.

link: http://mentalfloss.com/article/30363/why-does-your-nose-get-stuffy-one-nostril-time